Dynamics of Delay Equations, Theory and Applications - Abstract

Vas, Gabriella

Large-amplitude periodic solutions for delay equations with positive feedback

This talk studies scalar delay differential equations of the form [ dotxleft(tright)=-xleft(tright)+fleft(xleft(t-1right)right), ] where $f$ is a nondecreasing $C^1$-function. If $chi$ is a fixed point of $f$ with $f'left(chiright)>1$, then $left[-1,0right]ni smapstochiinmathbbR$ is an unstable equilibrium. We say that a periodic solution has large amplitude if it oscillates about at least two fixed points $chi_-<\chi_{+}$ of $f$ with $f'\left(\chi_{-}\right)>1$ and $f'left(chi_+right)>1$. We investigate what type of large-amplitude periodic solutions may exist at the same time when the number of such fixed points (and hence the number of unstable equilibria) is an arbitrary integer $Ngeq2$. It can be shown shown that the number of different configurations equals the number of ways in which $N$ symbols can be parenthesized. If time permits, we discuss the case $N=2$ in detail. For a class of nonlinearities $f$, the global attractor consists of two spindle-like structures (described in earlier works of T. Krisztin, H.-O. Walther and J. Wu) and the unstable sets of two large-amplitude periodic orbits. We can give a detailed geometric description for the unstable sets of these periodic orbits. Joint work with T. Krisztin.